- #1
lalbatros
- 1,256
- 2
From this reference:
titled From Classical to Quantum Mechanics,
I quote the following: ( [tex]\xi^i [/tex] are coordinate functions)
Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that
[tex]\{\xi^i,\xi^j\} \equiv \omega^i^j[/tex]
is an inversible matrix, we may define the inverse [tex]\omega_i_j[/tex] by requiring
[tex]\omega_i_j \omega^j^k = \delta_i^k[/tex]
We define a tensorial quantity
[tex]\omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j[/tex]
which turns out to be a non-degenerate 2-form.
This implies that the dimension of the manifold M is necessarily even.
My questions are the following:
I don't understand the two statement that I have put in red above.
What is a non-degenerate 2-form?
Why does this one above 'turns out' to be non-degenerate?
Why does that imply that M is even?
Additional comments would be welcome. Like concerning the meaning of [tex]\omega [/tex] above.
In addition, I guess the point here by a shorter way: I think that all odd-dimensional antisymmetric matrices are singular. Is there a link with the language used above?
Warm thanks in advance,
Michel
titled From Classical to Quantum Mechanics,
I quote the following: ( [tex]\xi^i [/tex] are coordinate functions)
Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that
[tex]\{\xi^i,\xi^j\} \equiv \omega^i^j[/tex]
is an inversible matrix, we may define the inverse [tex]\omega_i_j[/tex] by requiring
[tex]\omega_i_j \omega^j^k = \delta_i^k[/tex]
We define a tensorial quantity
[tex]\omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j[/tex]
which turns out to be a non-degenerate 2-form.
This implies that the dimension of the manifold M is necessarily even.
My questions are the following:
I don't understand the two statement that I have put in red above.
What is a non-degenerate 2-form?
Why does this one above 'turns out' to be non-degenerate?
Why does that imply that M is even?
Additional comments would be welcome. Like concerning the meaning of [tex]\omega [/tex] above.
In addition, I guess the point here by a shorter way: I think that all odd-dimensional antisymmetric matrices are singular. Is there a link with the language used above?
Warm thanks in advance,
Michel
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